Braided monoidal category

In mathematics, a commutativity constraint on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects .

A braided monoidal category is a monoidal category equipped with a braiding - that is, a commutativity constraint that satisfies the hexagon identities (see below). The term braided comes from the fact that the braid group plays an important role in the theory of braided monoidal categories. Partly for this reason, braided monoidal categories and various related notions are important in the theory of knot invariants.

Alternatively, a braided monoidal category can be seen as a tricategory with one 0-cell and one 1-cell.

For along with the commutativity constraint to be called a braided monoidal category, the following hexagonal diagrams must commute for all objects . Here is the associativity isomorphism coming from the monoidal structure on :

It can be shown that the natural isomorphism along with the maps coming from the monoidal structure on the category , satisfy various coherence conditions which state that various compositions of structure maps are equal. In particular:

The braiding commutes with the units. That is, the following diagram commutes:

The action of on an -fold tensor product factors through the braid group. In particular,

There are several variants of braided monoidal categories that are used in various contexts. See, for example, the expository paper of Savage (2009) for an explanation of symmetric and coboundary monoidal categories, and the book by Chari and Pressley (1995) for ribbon categories.

Sometimes categories are assumed to have n-ary monoidal products for all finite n (in particular n>2), diminishing the role of associator morphisms. In such categories, the following variant is used, where the hexagon axiom is replaced by the two conditions: